Excitons and their Fine Structure in Lead Halide Perovskite Nanocrystals from Atomistic GWBSE Calculations Giulia BifﬁyzkY eongsu ChokRoman Krahneyand Timothy C. Berkelbachx

2025-04-27 0 0 9.19MB 8 页 10玖币

Excitons and their Fine Structure in Lead Halide Perovskite

Nanocrystals from Atomistic GW/BSE Calculations

Giulia Bifﬁ,†,‡,kYeongsu Cho,¶,kRoman Krahne,†and Timothy C. Berkelbach∗,¶,§

†Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy

‡Dipartimento di Chimica e Chimica Industriale, Universit`a degli Studi di Genova, Via Dodecaneso, 31, 16146 Genova, Italy

¶Department of Chemistry, Columbia University, New York, New York 10027 USA

§Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010 USA

kThese authors contributed equally

E-mail: t.berkelbach@columbia.edu

Abstract

Atomistically detailed computational studies of nanocrystals, such

as those derived from the promising lead-halide perovskites, are

challenging due to the large number of atoms and lack of symme-

tries to exploit. Here, focusing on methylammonium lead iodide

nanocrystals, we combine a real-space tight binding model with the

GW approximation to the self-energy and obtain exciton wavefunc-

tions and absorption spectra via solutions of the associated Bethe-

Salpeter equation. We ﬁnd that the size dependence of carrier con-

ﬁnement, dielectric contrast, electron-hole exchange, and exciton

binding energies has a strong impact on the lowest excitation en-

ergy, which can be tuned by almost 1 eV over the diameter range of

2–6 nm. Our calculated excitation energies are about 0.2 eV higher

than experimentally measured photoluminescence, and they display

the same qualitative size dependence. Focusing on the ﬁne structure

of the band-edge excitons, we ﬁnd that the lowest-lying exciton is

spectroscopically dark and about 20–30 meV lower in energy than

the higher-lying triplet of bright states, whose degeneracy is slightly

broken by crystal ﬁeld eﬀects.

Introduction

Lead-halide perovskite nanocrystals (LHP NCs), with formula

APbX3where A is a monovalent cation and X is a halide anion,

exhibit remarkable electronic and optical properties,1–3suggest-

ing promising applications in optoelectronics,4–8photonics,9,10 and

spintronics.11 Compared to bulk LHPs, LHP NCs display anoma-

lously short radiative lifetimes and concomitant high photolumines-

cence quantum yields, the origin of which is still under debate.1–3,12

These interesting optical properties and their temperature depen-

dence have focused attention on the exciton ﬁne structure, i.e., the

energy ordering and character of the band-edge excitons.13?–19 For

example, the lowest-energy exciton of Cs-LHP NCs has been sug-

gested to be a bright (emissive) state with total angular momentum

J=1, which could explain the high photoluminescence quantum

yield.14–16 This property would be in stark contrast with the case

of conventional organic and inorganic semiconductors, where the

lowest-energy exciton is a dark (non-emissive) state. Other works

have suggested that this conventional behavior persists in the LHP

NCs (i.e., the lowest-energy exciton is a dark state with total an-

gular momentum J=0) but that exciton relaxation by phonons

is suppressed at low temperatures, yielding long lifetimes for the

bright exciton in both NCs17–19 and 2D LHPs.?

The properties of the band-edge excitons—whose energy sep-

arations are very small, of the order of 1–10 meV—are inﬂu-

enced by quantum conﬁnement, electron-hole attraction and ex-

change,14–16,20 dielectric environment,21 and lattice structure (in-

cluding the bulk and surface Rashba eﬀects),22–25 making precise

theoretical predictions an incredible challenge. This complexity

has led to the development of exciton models based on eﬀective-

mass or k·pHamiltonians, pioneered especially by Efros and co-

workers,14–16 among others.20,26 Atomistic simulation, although de-

sirable, is frustrated by the sizes of even the smallest experimentally

accessible NCs, which can have hundreds or thousands of atoms.

Although atomistic force-ﬁelds or density functional theory can be

applied to study the ground-state and structural properties of sys-

tems of this size,27–30 accurate excited-state theories are more ex-

pensive and generally inaccessible in a fully ab initio setting.

Here, we overcome this challenge and construct an atom-

istic orbital-dependent tight-binding model parametrized by ﬁrst-

principles density functional theory (DFT) calculations, after which

we use a model dielectric function to apply self-energy corrections

via the GW approximation31–33 and calculate the energies and prop-

erties of excitons via the Bethe-Salpeter equation.34–36 The com-

putational approach is similar to previous work by two of us on

the properties of layered quasi-two-dimensional LHPs.37 As a spe-

ciﬁc example, we study methylammonium lead iodide (MAPbI3)

NCs, calculating the exciton binding energies, absorption spectra,

and band-edge ﬁne structure. Within the approximations of our

approach, we ﬁnd that the lowest-energy exciton is always a dark

state.

We study MAPbI3for several reasons, although our approach

is general and could be applied to any LHP NC. First, this LHP

is one of the most studied in its bulk form, especially for pho-

tovoltaics, partly due to its strong absorption at low energies that

facilitates sensitization.38,39 Moreover, bulk MAPbI3crystals have

low lasing thresholds40 and are generally more stable than Cs-based

ones.41 For these reasons, the atomic and electronic structure of

bulk MAPbI3has been extensively studied using DFT42,43 and the

GW approximation,44–47 providing important points of comparison.

These valuable properties have motivated experimental studies of

MAPbX3NCs with high photoluminescence quantum yields and

controllable size, leading to tunable band gaps. Such MAPbX3

NCs have been realized by colloidal synthesis48–53 and templated

growth inside porous oxide ﬁlms,54,55 providing experimental re-

sults to which we can directly compare.

arXiv:2210.01324v1 [cond-mat.mtrl-sci] 4 Oct 2022

Computational Methods

The parameters of our orbital-dependent tight-binding model are

determined from a DFT calculation of bulk MAPbI3, which ﬁrst

requires the determination of the crystal structure. (To our knowl-

edge, the crystal structure of LHP NCs is still poorly understood,

but the crystal structure of MAPbI3NCs is likely tetragonal.1How-

ever, this assumption may be incorrect due to surface eﬀects and/or

metastability.) From experiments, it is known that the bulk MAPbI3

exists in an orthorhombic phase below about 160 K, a tetragonal

phase between 160 K and 330 K, and a cubic phase above 330 K.56

The computational prediction of crystal structures in ﬁnite temper-

ature phases is nontrivial because of anharmonic eﬀects. In Ref.

43, it was shown that using zero-temperature DFT for the geometry

optimization of the tetragonal and cubic phases yields a large inver-

sion symmetry breaking via distortion of the PbI framework, lead-

ing to a large bulk Rashba eﬀect. However, these crystal structures

disagree with second harmonic generation rotational anisotropy ex-

periments. Therefore, with interest in room-temperature behavior,

we reuse the structure proposed in Ref. 43, which was generated

by ﬁxing the PbI framework of the tetragonal phase and relaxing

only the MA cations, whose orientations yield a small but nonzero

Rashba eﬀect. This crystal structure is shown in Fig. 1(a).

With our MA-relaxed crystal structure of bulk MAPbI3, we per-

form a DFT calculation including spin-orbit coupling, using Quan-

tum Espresso57 with the PBE exchange-correlation functional,58

relativistic PAW pseudopotentials, a kinetic energy cutoﬀof 60 Ry,

and a 6 ×6×6k-point mesh. Using wannier90,59 the DFT so-

lution is used to construct maximally localized Wannier functions

(MLWFs) φµ(r), corresponding to I 5s and 5p orbitals and Pb 6s

and 6p orbitals, as well as their tight-binding Hamiltonian matrix

elements hµν and dipole matrix elements re

µν,e∈ {x,y,z}. Greek

indices µ, ν, κ, λ will be used throughout to indicate MLWF atomic

orbitals, including spin. These real-space tight-binding parameters

are then used to build a mean-ﬁeld Hamiltonian hNC of the aperi-

odic NCs. Our NC surfaces are terminated with halide atoms, in

agreement with experimental observations,49 with exposed facets

that are qualitatively equivalent to the (100) surface of a cubic per-

ovskite. We use the same tight-binding parameters for all atoms,

even those near the surface of the NCs, which can be understood as

an approximate passivation that does not require microscopic spec-

iﬁcation. However, the details of surface passivation are important

for controlling trap states and associated non-radiative recombina-

tion.28,51,53 Similarly, relaxation of the atoms at the surface, which

we neglect here, would modify these matrix elements and the re-

sulting electronic structure.25,28

With this tight-binding Hamiltonian matrix in the basis of

MWLFs φµ(r), we calculate the molecular orbitals Cµpand orbital

energies εp,hNCC=Cε. To the orbital energies, we add two self-

energy corrections in the spirit of the many-body GW approxima-

tion (at the level of approximation made throughout this work, we

do not distinguish degrees of self-consistency in the self-energy, but

our calculations are all performed in the one-shot G0W0approxi-

mation as described here). The ﬁrst correction is a rigid shift of the

conduction and valence band energies by ±∆/2 to correct the bulk

band gap to the previously calculated value of 1.67 eV.44 Our DFT

and GW band structures of bulk tetragonal MAPbI3are shown in

Fig. 1, which can be seen to contain a small but nonzero Rashba

splitting at the band edges. As is well-known,44–47 the GW correc-

tion to the band gap of LHPs is sizable—about 1.5 eV in our case.

Our second self-energy correction within the GW approximation

uses a model dielectric function and a static screening approxima-

tion to account for dielectric constrast at the NC interface,60,61 in

the style of a GδW calculation.62 Combining these two corrections,

we calculate the GW quasiparticle energies of all conduction (c)

(a) (b)

(c)

Energy (eV)

DFT

Z A M ΓZ R X Γ

Figure 1: (a,b) Crystal structure of bulk tetragonal MAPbI3from

Ref. 43 and (c) band structure calculated with DFT (PBE with spin-

orbit coupling) and rigidly shifted to match the GW band gap.44

and valence (v) bands,

Ec=εc+ ∆/2+X

|Cµc|2δΣ(rµ) (1a)

Ev=εv−∆/2−X

|Cµv|2δΣ(rµ) (1b)

where rµis the position of the atom to which the MLWF φµ(r)

belongs,

δΣ(r1)=1

2lim

r2→r1"W(r1,r2)−1

p|r1−r2|#,(2)

and W(r1,r2) is the classical Coulomb interaction energy be-

tween two charges in the NC at r1,r2.60 We use the analytical

Coulomb energy of charges in a dielectric cuboid with side lengths

Lx,Ly,Lz,63

W(r1,r2)=

∞

i,j,k=−∞

[(p−env)/(p+env)]|i|+|j|+|k|

p|r1−ri jk

2|(3a)

ri jk

2≡[(−1)ix2+iLx,(−1)jy2+jLy,(−1)kz2+kLz].(3b)

With the surface termination described above, For a NC made of

n3octahedra, we use Lx=Ly=n×(6.4 Å) +2×(1.5 Å) and

Lz=n×(6.5 Å) +2×(1.5 Å), where 6.4 Å and 6.5 Å correspond

to the diagonals of the tetragonal octahedra and we have added an

additional dielectric buﬀer region of 1.5 Å on all sides. Through-

out this work, we use the bulk MAPbI3perovskite high-frequency

dielectric constant p=6.1 and an environmental dielectric con-

stant of εenv =2.1, representative of a generic low-dielectric en-

vironment, such as organic ligands.2,8,64 (For large NCs with small

exciton binding energies that are on the order of optical phonon fre-

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ExcitonsandtheirFineStructureinLeadHalidePerovskiteNanocrystalsfromAtomisticGW/BSECalculationsGiuliaBif,y,z,kYeongsuCho,{,kRomanKrahne,yandTimothyC.Berkelbach,{,xyIstitutoItalianodiTecnologia,ViaMorego30,16163Genova,ItalyzDipartimentodiChimicaeChimicaIndustriale,UniversitadegliStudidiGenova,ViaDo...

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Excitons and their Fine Structure in Lead Halide Perovskite Nanocrystals from Atomistic GWBSE Calculations Giulia BifﬁyzkY eongsu ChokRoman Krahneyand Timothy C. Berkelbachx

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